Wettability control of droplet durotaxis

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Citation Bueno, Jesus et al. “Wettability Control of Droplet Durotaxis.” SoftMatter 14, 8 (2018): 1417–1426 © 2018 Royal Society of Chemistry

As Published http://dx.doi.org/10.1039/c7sm01917c

Publisher Royal Society of Chemistry

Version Author's final manuscript

Citable link http://hdl.handle.net/1721.1/120323

Terms of Use Creative Commons Attribution-Noncommercial-Share Alike

Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/

Wettability control of droplet durotaxisJesus Bueno1,*, Yuri Bazilevs2, Ruben Juanes3, and Hector Gomez4

1Departamento de Matematicas, Universidade da Coruna. Campus de Elvina, A Coruna, 15192, Spain2Department of Structural Engineering, University of California, San Diego. 9500 Gilman Drive, La Jolla, CA 92093,USA3Department of Civil and Environmental Engineering, Massachusetts Institute of Technology. 77 MassachusettsAvenue, Cambridge, MA 02139, USA4School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN 47907, USA*jesus.bueno@udc.es

ABSTRACT

Durotaxis refers to cell motion directed by stiffness gradients of an underlying substrate. Recent work has shown that dropletsalso move spontaneously along stiffness gradients through a process reminiscent of durotaxis. Wetting droplets, however,move toward softer substrates, an observation seemingly at odds with cell motion. Here, we extend our understanding of thisphenomenon, and show that wettability of the substrate plays a critical role: while wetting droplets move in the direction of lowerstiffness1, nonwetting liquids reverse droplet durotaxis. Our numerical experiments also reveal that Laplace pressure can beused to determine the direction of motion of liquid slugs in confined environments. Our results suggest new ways of controllingdroplet dynamics at small scales, which can open the door to enhanced bubble and droplet logic in microfluidic platforms.

Introduction

The ability to generate and manipulate liquid droplets as well as to control how they interact with solid substrates has attractedincreased interest in the scientific community due to the number of applications in which droplet dynamics plays a fundamentalrole. Understanding, predicting and controlling these processes is essential, for example, in the design of new materials anddevices at small scales2–5; droplets are present in optofluidic optical attenuators6, microfluidic electronic paper7 and, in general,in bubble- and droplet-based microfluidic platforms8–10 that are used for industrial, biological and chemical applications, suchas high- and ultrahigh-throughput screening11–13, enzymatic assays14 and chemical synthesis15.

During the past few decades, important advances have been achieved in the study of droplet motion in solid substrates16–18.Different mechanisms have been identified and explored to overcome the contact angle hysteresis and induce droplet motion,ranging from the use of chemical, thermal and electrical gradients19–21 to the manipulation of the surface topography22 or theapplication of external vibrations23. However, despite significant progress in recent years, droplet motion—especially in relationto deformable substrates—is not fully understood. New mechanisms to control droplet motion on deformable solids have beenproposed recently. For example, droplet motion may be driven by gradients in strain of the substrate24—a process termedtensotaxis by analogy with the behavior previously observed in cells25. Another cell motion mechanism—durotaxis—hadalready inspired new ways of controlling droplet dynamics: wetting droplets deposited on substrates with nonuniform stiffnesstend to move toward the softer parts of the substrate1. This directed motion, however, is seemingly opposite to the behaviorobserved in cells25.

As it turns out, a central element of droplet durotaxis has heretofore remained unexplored. Here, we show that the behaviorof droplets in substrates with nonuniform stiffness depends critically on the wettability of the substrate with respect to theliquid: while wetting droplets move toward the softer parts of the substrate—in agreement with the experiments reported byStyle et al.1—nonwetting droplets exhibit the opposite behavior and move in the direction of greater stiffness—similarly tothe response of cells to rigidity gradients of the underlying substrate25. We also elucidate the role of Laplace pressure in thismechanism, and our results suggest that Laplace pressure may be used to control the direction of motion of nonwetting droplets.

This improved understanding of the impact of wetting on droplet durotaxis stems from a computational model of theinteraction of deformable solids and multiphase fluids, which accounts for the dynamically coupled, nonlinear problem inthree dimensions. The proposed theory allows to make predictions via high-fidelity numerical simulations that elucidate thephenomenon and inform future experimental designs.

Model of droplet durotaxisWhen a small liquid droplet is deposited on a flat, rigid and chemically homogeneous solid surface, the droplet tends to adopta spherical shape and modify its contact angle until it reaches the mechanical equilibrium. The value of the static contactangle α depends on the surface tension at the liquid–vapor γLV , solid–vapor γSV and solid–liquid γSL interfaces, and can beapproximated by taking the following balance of forces at the contact line [see Fig. 1(A)],

γSL + γLV cos(α) = γSV . (1)

This expression, known as the Young–Dupre equation, has been known for almost two centuries. Equation (1), however, is onlyvalid for ideal solid surfaces, where the solid is flat and infinitely rigid. When the Laplace pressure or the surface tension at theliquid–vapor interface are sufficiently strong so as to deform the substrate, the Young–Drupre equation is no longer valid; seeFig. 1. This may happen for slender structures26–28 but also when the droplet is small or when it is deposited on a sufficientlysoft substrate29, 30. The elastocapillary length scale LEC ∼ γLV/E, where E is the substrate’s Young modulus, allows to estimatewhen the elastocapillary forces are relevant and the Young–Dupre equation breaks down.

For most solids and typical values of liquid surface tension, this results in immeasurably small elastocapillary lengthscales that are much smaller than the radius of the droplet. In these cases, the deformation of the solid is negligible and theYoung-Dupre theory is valid. For soft solids such as some gels, however, LEC can approach or even exceed the size of wettingdroplets. In such cases, in which small droplets wet soft substrates, the interfacial forces create a ridge at the contact line andthe Laplace pressure dimples the substrate under the droplet; see Fig. 1(B). As a consequence, there is a rotation of the contactline30 and the apparent contact angle, that is, the angle formed by the liquid–vapor interface and the undeformed surface ofthe substrate (ϕ in Fig. 1), differs from the angle predicted by the Young–Dupre equation. When a droplet is deposited on asubstrate with variable stiffness, the rotation of the contact line—and thus, the apparent contact angle—is different at each sideof the droplet, resulting in an imbalance of horizontal forces that may trigger the motion of the droplet.

Figure 1. Wetting on rigid and deformable substrates at small scales. (A) Liquid droplet (blue) deposited on a rigid substrate(dark gray). The spherical shape adopted by the droplet depends on the surface tensions at the contact line: γLV , γSV and γSL.The static contact angle α is given by the Young–Dupre equation. (B) Liquid droplet on a soft substrate (light gray). Thesurface tension at the liquid–vapor interface γLV creates a ridge at the contact line and the Laplace pressure ∆p dimples thesubstrate under the droplet. The contact lines rotate and thus, the apparent contact angle ϕ differs from the Young–Duprecontact angle α .

Previous studies of the interaction of droplets and deformable substrates29, 31 are based on linear elastic solids and thin filmdescriptions of the fluid that allow for the computation of minimum-energy configurations. To advance our understanding ofthe impact of wettability—especially in the regime of nonwetting droplets—here we propose to model the interaction betweendroplets and deformable solids using a three-dimensional theory that couples a two-phase fluid with a nonlinear solid. Thefluid is composed of a liquid and a gaseous phase separated by a diffuse interface that accurately captures surface tension32.The proposed model is solved computationally using a spline-based finite-element method known as isogeometric analysis33,ideally suited for the simulation of high-order partial differential equations that arise as a result of interfacial effects.

Results

Wettability and droplet durotaxisThe dynamics of wetting droplets (α < 90◦) on substrates with nonuniform stiffness has been studied previously30. However,the role that liquid wettability plays in droplet durotaxis remains unexplored. To study this problem we start by mimicking theexperiments conducted by Style et al.30. We model a solid substrate (see Fig. 2) composed of two different layers; the lowerlayer is a rigid material (dark gray) and the upper layer is a deformable solid (light gray). Both materials extend along thesolid substrate but their thickness varies in one of the horizontal directions so that the total thickness of the composite substrate

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Figure 2. Nonwetting droplets move in the direction of higher stiffness. A liquid droplet (blue) that forms a contact angle ofα = 120◦ with the solid surface is deposited on a substrate with nonuniform stiffness. The substrate is composed of a soft (lightgray) and a rigid material (dark gray). The initial position of the droplet is indicated by a semitransparent spherical cap. Thedroplet moves to the right, where the thickness of the soft material is smaller. We plot the streamlines of the fluid velocity alongthe mid-plane, which are colored with the velocity magnitude. The computational domain is a box of size 1.0×0.8×0.5. Thecomputational mesh is comprised of 100×80×50 C 1-quadratic elements. We have used the parameters ν = 0.125,µ = 1/200, γ = 2/100, and θ = 0.39. The stiffness of the substrate is E = 0.42 in the soft material and E = 124.2 in the rigidmaterial. The radius of the droplet is R = 0.11. See Methods section for the definition of these quantities.

remains constant. The rigid layer has a lenticular shape (see inset in Fig. 2) and is coated with the deformable solid, creatinga flat solid surface. This configuration results in a substrate that has a nonuniform rigidity. The solid is stiffer in the regionswhere the thickness of the soft material is smaller.

We place a nonwetting droplet (α = 120◦) on one of the softer regions of the substrate and let the droplet evolve freely inthe absence of gravity/external forces. In Fig. 2 we show the configuration of the liquid droplet at time t = 181 (the isosurfaceof dark blue color represents the liquid–vapor interface of the droplet). The initial position of the droplet is indicated insemitransparent light blue color. We also plot the streamlines of the fluid velocity along the mid-plane. The streamlines arecolored with the velocity magnitude. The numerical experiment in Fig. 2 shows an important result. The nonwetting dropletadvances in the direction of higher stiffness with a time-decreasing velocity; in contrast with what had been observed for wettingdroplets. This observation shows that the direction of droplet durotaxis may be reversed by simply altering the wettability of theliquid. To further study the mechanisms of durotaxis, in what follows we adopt a simplified configuration in two dimensionsand with solid substrates of constant rigidity gradients.

Mechanistic underpinning of droplet durotaxisWe start by comparing the behavior of wetting and nonwetting droplets on deformable substrates with nonuniform stiffness.We use our nonlinear fluid–structure interaction model to simulate a solid substrate (gray in Fig. 3) with spatially variablestiffness dynamically coupled to a two-phase fluid with surface tension. The rigidity of the substrate follows a linear profileand is lower on the left boundary (light gray) and higher on the right boundary (dark gray) of the solid domain. We place

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Figure 3. Droplet motion driven by durotaxis. (A) and (B) show the initial configurations: a liquid droplet (blue) is depositedon a substrate (gray) with nonuniform stiffness. On the left column, the droplet initially forms a contact angle of α = 60◦ withthe substrate. On the right column, the droplet is a nonwetting liquid with α = 120◦. In both cases, the rigidity of the solidfollows a linear profile reaching a maximum at the right end. Figs. (C) and (E) show the wetting droplet at two differentinstants. Figs. (D) and (F) present the time evolution of the nonwetting droplet. The initial position of the droplets is illustratedwith a black solid line that represents the liquid–vapor interface at t = 0. Figs. (C)–(F) also show the streamlines of the fluidvelocity colored by velocity magnitude. The results show that wetting droplets move toward softer parts of the substrate with anincreasing velocity magnitude [(C), (E)]. Nonwetting droplets move toward stiffer areas with a decreasing velocity magnitude[(D), (F)]. The computational domain is a box of size 1.0×0.5, discretized with a mesh of 256×128 C 2 elements. Thesubstrate thickness is 0.15 and its Poisson ratio ν = 0.125. The maximum value of the Young modulus is Emax = 2.75. Theminimum value is Emin = 0.1 for the wetting droplet and Emin = 0.01 for the non-wetting liquid. For the fluid, we have adoptedthe parameters, µ = 1/512, γ = 2/256, and θ = 0.39. The radius of the droplets is R = 0.13.

a liquid droplet (blue) on the solid surface and let it move freely in absence of external forces. We analyze the behavior fortwo different Young–Dupre contact angles. Figs. 3(A) and (B) show the initial configuration of the droplets. The left columnin Fig. 3 illustrates the time evolution of a wetting droplet (α = 60◦), and the right column the corresponding evolution of anonwetting droplet (α = 120◦).

The results of our simulations show a remarkable difference in the behavior of wetting and nonwetting droplets, consistentwith the three-dimensional results of Fig. 2. While the wetting droplet moves toward the softer part of the substrate, thenonwetting droplet advances in the opposite direction, i.e., it migrates up rigidity gradients. Similar numerical experimentswere carried out for other linear stiffness profiles and other values of the Young–Drupre contact angle (data not shown). Theresults unveiled that the value of the contact angle α for which the direction of the motion is reversed depends on the rigidityof the substrate for a given liquid droplet and Poisson ratio. For the cases that we have analyzed, the motion is reversed forα ∈ [105◦,108◦], where the larger values correspond to stiffer substrates. We have also observed that this value increases withthe Poisson ratio.

The streamlines of the fluid velocity, colored by velocity magnitude in Fig. 3, show that droplet motion is able to create avortical structure in the vapor phase. For the wetting droplet (left column), the magnitude of the droplet velocity increaseswith time [Fig. 4(A)]. For the nonwetting droplet (right column), the velocity magnitude decreases with time [Fig. 4(B)]. Thisindicates that droplets move faster in softer areas of substrates with constant stiffness gradients.

The mechanistic explanation for the observed droplet velocity rests on the apparent contact angle at the front and rear

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Figure 4. Time evolution of the droplet velocity and the apparent contact angles for the wetting and nonwetting dropletsshown in Fig. 3. (A) The wetting droplet moves faster as it advances toward the softer part of the substrate. (B) The nonwettingdroplet decelerates as it advances toward the stiffer region. The velocity results are consistent with the evolution of the apparentcontact angle difference at both sides of the droplets. (C) and (D) show the time evolution of the apparent contact angles at thefront and the rear contact lines. (C) For the wetting droplet, the apparent contact angle on the left (front) side is smaller than thecontact angle on the right (rear) contact line. The difference between both contact angles increases as the droplet moves towardthe softer area. (D) The nonwetting droplet also moves in the direction of the smaller apparent contact angle (right contact line).As the droplet moves toward the stiffer area, the difference between the front and the rear contact angles decreases.

contact lines of the droplets (relative to the displacement direction); see Figs. 4(C) and (D) for the wetting and nonwettingdroplets, respectively. In both cases, the droplet moves toward the contact line in which the apparent contact angle is smaller,that is, the “softer” contact line in wetting droplets and the “stiffer” contact line in nonwetting droplets. For a wetting droplet,the difference between the two apparent contact angles increases with time, as does the velocity magnitude [Fig. 4(A) and (C)].In contrast, for a nonwetting droplet, the difference between the apparent contact angles at each side of the droplet decreaseswith time. As the droplet moves to the stiffer region, the surface tension and the Laplace pressure produce a smaller deformationof the substrate and a smaller rotation of the contact line. As a result, the apparent contact angle approaches the static contactangle predicted by Young–Dupre equation. When both rear and front apparent contact angles reach the same value, the dropletstops. These results indicate that the dynamics of the droplets seem to be primarily controlled by the apparent contact angles atopposite sides of the droplet.

The wetting and nonwetting droplets analyzed in these simulations have the same radius and surface tension and, thus,the Laplace pressure is also the same in both cases. If we assume that the droplets have the shape of a circular segment, thedistance between the two contact lines will be the same in both cases because the wetting and nonwetting contact angles aresupplementary. Since the Laplace pressure takes the same value in both cases and acts on the same length, the contact linerotation produced by the Laplace pressure has the same sign for the wetting and the nonwetting droplet, that is, clockwise forthe left contact line and counter-clockwise for the right contact line. Note also that the apparent contact angle is smaller thanthe Young–Drupre contact angle for wetting droplets [Fig. 4(C)]. For nonwetting droplets, the opposite is true [Fig. 4(D)].This indicates that the rotation of the contact lines produced by surface tension has different sign for wetting and nonwettingdroplets. For example, in the left contact line the rotation seems to be clockwise for wetting droplets and counter-clockwise fornonwetting droplets. This suggests that the rotation induced by the surface tension in nonwetting droplets has opposite sign andgreater absolute value than the rotation caused by the Laplace pressure.

To confirm this hypothesis we carried out several computations on substrates with uniform stiffness for a given droplet

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Figure 5. Apparent contact angle ϕ with respect to the contact angle α predicted by Young–Dupre equation for a given liquiddroplet and for substrates of uniform stiffness. In rigid substrates, the deformation produced by surface tension and Laplacepressure is negligible and the apparent contact angle matches the Young–Dupre equilibrium contact angle (black solid line).For sufficiently soft substrates, the Laplace pressure ∆p and the surface tension γLV deform the solid, inducing a rotation φ ofthe contact line (gray solid line). When ϕ = 90◦, surface tension points vertical and induces no rotation of the contact line, i.e.,φγ = 0. The rotation is entirely produced by the Laplace pressure, i.e., φ = φ∆p, which is constant for all the computations withthe same substrate stiffness (blue dotted area). The remaining contribution to the rotation (red striped area) is attributed tosurface tension. Note that for ϕ ' 105◦, φγ =−φ∆p and the apparent contact angle matches the Young–Dupre contact angle.For ϕ & 105◦, | φγ |≥| φ∆p | and the final apparent contact angle exceeds the contact angle predicted by the Young–Dupreequation. The computations were performed on a box of size 1.0×0.5 and using 256×128 C 2 elements. The droplet radius isR = 0.13 and the substrate thickness 0.15. We have used the parameters ν = 0.125, Eso f t = 0.2, µ = 1/512, γ = 2/256, andθ = 0.39.

radius and surface tension; see Fig. 5. We measured the steady-state value of the apparent contact angle ϕ on a soft substrate(gray solid line) for different Young–Dupre contact angles α . We compared our measurements with the values of ϕ on ainfinitely rigid solid (black solid line). Let us call φγ and φ∆p the contact line rotations produced by surface tension and Laplacepressure, respectively. The total rotation of the contact line is φ = φγ +φ∆p. The results in Fig. 5 confirm that the rotation of thecontact line, φ , changes sign approximately when α ' 105◦. For ϕ & 105◦, φγ and φ∆p have opposite sign and | φγ |>| φ∆p |.These observations suggest that a relative increase of the Laplace pressure with respect to surface tension could potentiallychange the sign of the rotation at the contact lines and thus reverse the direction of droplet motion for nonwetting droplets.

Controlling durotaxis by confinementTo study the role of Laplace pressure in droplet motion driven by durotaxis, we propose a system in which this quantity can beeasily manipulated. We place a liquid slug in between two identical planar surfaces forming a capillary bridge (see Fig. 6).In this system, the Laplace pressure scales with the distance between the two solid surfaces. Thus, one can alter the relativestrength of the Laplace pressure with respect to surface tension by changing the distance between the solids while keeping theYoung–Drupre contact angle α constant.

We carry out two different simulations; see Figs. 6(A) and (B) for the initial configurations. In both cases, the solids havea nonuniform stiffness that follows a linear profile. They are softer on the left boundary (light gray) and stiffer on the rightboundary (dark gray). Each simulation has a different distance between the two solids, while the Young–Dupre contact angle isthe same for both cases, α = 120◦, corresponding to a nonwetting liquid slug. For the larger separation between substrates (leftcolumn of Fig. 6), the Laplace pressure is lower. In this case, the liquid moves toward the stiffer parts of the substrate. Thedynamics seems to be controlled by surface tension. The time evolution of the difference between the apparent contact anglesat both sides of the liquid bridge is similar to that observed for the nonwetting droplet analyzed in Fig. 4(D). When the gapbetween substrates is 10 times smaller (right column of Fig. 6), the Laplace pressure is 10 times larger. In contrast with the

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Figure 6. Laplace pressure can be used to alter the direction of confined liquid slugs. A liquid droplet (blue) is placed inbetween two identical solids (gray) forming a capillary bridge. The solids’ stiffness increases linearly from left to right. Weanalyze two different cases for the same Young-Drupre contact angle (α = 120◦) and the same surface tension. We do modifythe distance between the solids. On the left column the gap is bigger, which results in a lower Laplace pressure. (A) and (B)represent the initial configurations of the problem. (C)-(F) depict the configuration of the capillary bridges at different timeinstants. The results show that for large Laplace pressures (right column) the droplet moves toward softer regions. For lowLaplace pressure, the liquid moves toward stiffer areas. The computational domain is a box of size 1.0×0.5 for the leftproblem and a box of size 1.0×0.25 for the right problem. They are both discretized with 256×128 C 1 quadratic elements.We use the parameters µ = 1/512, γ = 2/256, θ = 0.39 and ν = 0.125. The rigidity of the substrate varies linearly betweenEmin = 0.34 and Emax = 8.28. The thickness of the solids is 0.15 and 0.115 for the left and the right problems, respectively.Note that for visualization purposes in the right problem we are only showing the central part of the domain.

previous case, here the capillary bridge moves toward the softer part of the solid, showing that durotaxis of liquid slugs can bereversed by judiciously manipulating the Laplace pressure in confined environments.

DiscussionIt is generally accepted that, when a droplet is deposited on deformable substrates, the surface tension at the liquid–vaporinterface and the Laplace pressure of the droplet produce a rotation of the contact lines29–31, 34. The numerical experimentsconducted here show that this rotation is remarkably different for wetting and nonwetting droplets. While in wetting dropletsthe rotation induced by the surface tension and the Laplace pressure have the same sign, the opposite is true for nonwettingdroplets: the rotation produced by surface tension may outweigh the effect of the Laplace pressure, resulting in apparent contactangles that are larger than the Young–Drupre contact angle. Our results also indicate that the total rotation of the contact line islarger in softer substrates regardless of the wettability of the liquid. These observations explain why in our simulations theapparent contact angle is smaller in the softer part of the substrate for wetting droplets, while the opposite is observed fornonwetting droplets.

Our results support the theory proposed by Style et al.1 which suggests that–analogously to droplet motion driven byinterfacial energy gradients24—droplet durotaxis is to a large extent controlled by the difference between the apparent contact

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angles at the rear and the front contact lines of the droplet. Droplets move in the direction of the contact line that presents asmaller apparent contact angle. We have shown that for nonwetting droplets this may produce motion toward stiffer areas,mimicking the behavior of cell durotaxis. For wetting droplets, the contact angle difference and the droplet velocity bothincrease as the droplet moves toward softer areas of the substrate. In contrast, for nonwetting droplets, the contact angledifference and the velocity both decrease as droplets advance toward stiffer areas. Droplets move faster in the softer areas of thesubstrate for constant stiffness gradients.

Our study opens new possibilities to control the dynamics of droplets in soft substrates. The Laplace pressure may be usedto reverse the direction of durotaxis for nonwetting liquid slugs under confinement. Droplet motion can also be controlled bysimply altering the total wettability of the liquid, which can be accomplished, e.g., by using surfactants.

Several aspects of droplet durotaxis are still unexplored and should be the focus of future research. The interaction ofmultiple droplets in substrates with nonuniform stiffness or the influence of variable stiffness gradients in droplet velocity couldunveil interesting behavior for a better understanding of this phenomenon. Droplet durotaxis appears to be a powerful way ofcontrolling not only the velocity but the direction of droplet motion.

MethodsWe developed a model for the interaction of liquid droplets and deformable substrates, similar to the one presented in Bueno etal.35. The model captures the coupling between a nonlinear hyperelastic solid and a multiphase fluid that permits the stablecoexistence of a liquid and a gaseous phase separated by a diffuse interface endowed with surface tension.

Solid mechanicsWe use the momentum balance equation to describe the behavior of the solid, which in Lagrangian form can be expressed as

ρs0

∂ 2uuu∂ t2

∣∣∣∣XXX= ∇XXX ·PPP. (2)

Here, ∇XXX is the gradient with respect to the material coordinates XXX and |XXX indicates that the time derivative is taken by holdingXXX fixed; uuu is the solid displacement and ρs

0 is the mass density in the initial configuration; PPP is the first Piola–Kirchhoff stresstensor. As constitutive theory we adopt a nonlinear hyperelastic material. We use the generalized neo-Hookean model withdilatational penalty proposed by Simo and Hughes36. In this model, the second Piola–Kirchhoff stress tensor can be defined as

SSS = µJ−2/d(

III− 1d

tr(CCC)CCC−1)+

κ

2(J2−1

)CCC−1, (3)

where III represents the identity tensor, d denotes the number of spatial dimensions and tr(·) stands for the trace operator; κ

and µ are the material bulk and shear moduli, which are expressed as a function of the Young modulus E and the Poissonratio ν using the relations κ = E/(3(1−2ν)) and µ = E/(2(1+ν)); J is the determinant of the deformation gradient, that is,J = det(FFF), where FFF = III +∇X uuu. CCC denotes the Cauchy–Green deformation tensor, i.e., CCC = FFFT FFF . The first Piola–Kirchhoffstress tensor is obtained by taking PPP = FFFSSS. The solid Cauchy stress tensor can be computed using σσσ s = J−1FFFSSSFFFT = J−1PPPFFFT .

Fluid mechanicsThe behavior of the fluid is described by the isothermal form of the Navier–Stokes–Korteweg (NSK) equations. The NSKsystem constitutes the most widely accepted theory for the description of single-component two-phase flows and it naturallyallows for phase transformations in the fluid due to pressure and/or temperature variations. Since we are only interested instudying droplet motion on solid substrates, we select a parameter regime in which mass transfer between the liquid and gaseousphase is negligible. Our approach is based on the diffuse-interface or phase-field method32, i.e., an alternative to sharp-interfacemodels in which interfaces are replaced by thin transition regions. This allows an efficient computational treatment of thecoupled multiphysics problem. In the Eulerian description, the NSK equations can be expressed as

∂ρ

∂ t+∇ · (ρυυυ) = 0, (4a)

∂ (ρυυυ)

∂ t+∇ · (ρυυυ⊗υυυ)−∇ ·σσσ f = 0, (4b)

where ⊗ represents the outer vector product, ρ is the fluid density and υυυ denotes the velocity vector. σσσ f is the Cauchy stresstensor of the fluid, i.e., σσσ f = τττ− pIII + ςςς . Here, τττ stands for the viscous stress tensor, p denotes the pressure, and ςςς is known asthe Korteweg tensor. We consider Newtonian fluids, so the viscous stress tensor takes the form

τττ = µ(∇υυυ +∇

Tυυυ)+ λ∇ ·υυυIII, (5)

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where µ and λ are the viscosity coefficients. We assume that the Stokes hypothesis is satisfied, that is, λ =−2µ/3. In order toallow for the stable coexistence of liquid and gas phases we derive the thermodynamic pressure from the Helmholtz free-energyof a van der Waals fluid37, 38. The resulting van der Waals equation is expressed as

p = Rb(

ρθ

b−ρ

)−aρ

2, (6)

and gives the pressure p in terms of density and temperature θ . a and b are positive constants and R is the specific gas constant.The Korteweg tensor39, 40 is defined by

ςςς = λ

(ρ∆ρ +

12|∇ρ|2

)III−λ∇ρ⊗∇ρ. (7)

where λ > 0 is the capillarity coefficient and | · | denotes the Euclidean norm of a vector. The Korteweg tensor gives rise to thecapillary forces that are withstood by the liquid–vapor interfaces.

For the fluid problem we adopt the classical solid-wall boundary conditions. Additionally, the third-order partial spatialderivatives of the fluid equations require an extra condition to render a well-posed boundary value problem. To do so, weimpose ∇ρ ·nnn f = |∇ρ|cosα , where nnn f is the unit outward normal to the fluid boundary, and α is the contact angle between theliquid–vapor interface and the solid surface (see Fig. 1). Note that this boundary condition allows the imposition of the contactangle α at the fluid–structure interface. The apparent contact angle ϕ , however, is determined as part of the solution of theproblem.

Coupled problemThe partial differential equations and boundary conditions associated with the fluid and structure problems must be satisfiedsimultaneously. The two systems are coupled at the fluid-–structure interface in order to ensure compatible kinematics(υυυ = ∂uuu/∂ t) and transmission of tractions (σσσ f nnn f −σσσ snnn f = 0) between the fluid and solid domains.

Computational methodWe solve the coupled system consisting of Eq. (2) and Eq. (4) subject to the kinematic compatibility and traction balanceconstraints. Equation (2) is solved in the reference (undeformed) configuration of the solid domain. Equation (4) is solved in thespatial domain occupied by the fluid, which changes over time. This requires the use of geometrically flexible algorithms, such asthe finite element method. Here, we use isogeometric analysis, which is a spline-based finite-element-like method that combinesgeometric flexibility with smooth basis functions33, 41. The use of smooth basis functions allows for a direct discretizationof higher-order partial differential equations such as the NSK equation. To enable the use of classical finite-difference-typemethods for time integration, we recast the NSK equations in an arbitrary Lagrangian Eulerian (ALE) formulation:

∂ρ

∂ t

∣∣∣∣xxx+(υυυ− υυυ

)·∇ρ +ρ∇ ·υυυ = 0, (8a)

ρ∂υυυ

∂ t

∣∣∣∣xxx+ρ

(υυυ− υυυ

)·∇υυυ−∇ ·σσσ f = 0. (8b)

Here, υυυ is the fluid domain velocity and xxx is a coordinate in a reference domain that is used for computational purposes.Equations (2) and (8) can then be written in variational form and discretized in space using isogeometric analysis. We use thegeneralized–α method42 as a time integration scheme. The nonlinear system of equations is solved using a Newton–Raphsoniteration procedure, which leads to a two-stage predictor–multicorrector algorithm. The resulting linear system is solved usinga preconditioned GMRES method.

We express the problem in nondimensional form by rescaling the units of measurement of time, length, mass and temperatureby L0/

√ab, L0, bL3

0 and θc, respectively. Here, L0 = 1 is a length scale of the computational domain and θc = 8ab/(27R) isthe so-called critical temperature. Using this nondimensionalization, the problem can be characterized by five dimensionlessnumbers, i.e., the Poisson ratio ν and the dimensionless Young modulus E = E/(ρs

0ab) for the solid problem and thedimensionless surface tension γ =

√λ/a/L0, viscosity µ = µ/(L0b

√ab) and temperature θ = θ/θc for the fluid equations.

References1. Style, R. W. et al. Patterning droplets with durotaxis. Proc. Natl. Acad. Sci. 110, 12541–12544 (2013).

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AcknowledgementsHG was partially supported by the European Research Council through the FP7 Ideas Starting Grant Program (Contract#307201). HG and JB were partially supported by Xunta de Galicia, co-financed with FEDER funds. JB is grateful to thePh.D. student grant UDC-Inditex for the financial support during his visit at the University of California, San Diego, wherepart of this work was conducted. YB ... RJ acknowledges funding from the U.S. Department of Energy through a DOEMathematical Multifaceted Integrated Capability Center Award (Grant No. DE-SC0009286).

Author contributions statementHG and JB conceived the project and led the research process; JB developed the code and performed the numerical experiments;HG and JB carried out most of the theoretical analysis and wrote the manuscript. RJ and YB jointly contributed the ideas anddiscussed the results. All authors reviewed the manuscript.

Additional informationCompeting financial interests The authors declare that they have no competing interests.

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